(1) Field of the Invention
The present invention relates to a transducer-like device capable of clamping onto a tow cable. The device is responsive to the tow cable vibrating over the length of the cable such that the vibrating tow cable is an energy source for the device. A typical long cable that strums over a length supports vibrational energy produced by transverse waves of relatively short wavelengths and longitudinal energy produced by much longer longitudinal waves.
(2) Description of the Prior Art
Cable strum occurs because of vortex shedding from a cable towed at an angle with respect to the flow around the cable. A tow cable supports both transverse waves and longitudinal waves. The transverse waves tend to have short wavelengths: their propagation speed “c” (in meters per second) is approximately c=√{square root over (Tg/W)}; where “T” is the tension in Newtons; “W” is the weight per unit length and “g” is acceleration due to gravity, or 9.81 m/s2 (wherein “m” is meters and “s” is seconds). In water, W includes the weight of the added mass, which is equal to the weight of the displaced water.
Conversely, longitudinal waves have much longer wavelengths because of a propagation speed governed by c=√{square root over (E/ρ)}, where “E” and “ρ” are respectively the Young's modulus (having units of Newtons per square meter) and density (having units of kilograms per cubic meter) of the cable. As a result, on the order of one to ten longitudinal waves are contained by a mile-long tow cable.
Each transverse wave creates a localized region of curvature in the cable that shortens the cable (FIG. 1). The cable shortening generates longitudinal waves that exhibit twice the frequency of the transverse waves because the cable is shortened twice during each transverse wave cycle (FIG. 2). This is sometimes referred to as the frequency doubling effect.
The transverse wave frequency is governed by the formula for the Strouhal frequency, i.e., fs=0.2 U sin θ/d, where “U” is the tow speed, (in meters per second), “θ” is the incidence angle (in degrees) with respect to the flow, and “d” is the cable diameter (in inches).
Towed arrays typically include a steel cable that is approximately a mile long. The movement for such a cable is nearly straight over an entire length and at a critical angle. The critical angle is the angle at which the weight of the cable in water balances the drag of the cable. The critical angle is determined by the equation:
                                          1            2                    ⁢                      (                          σ              -              1                        )                    ⁢          π          ⁢                                          ⁢          gd          ⁢                                          ⁢          cos          ⁢                                          ⁢          θ                =                              (                                                            C                  D                                ⁢                sin                ⁢                                                                  ⁢                θ                            +                              π                ⁢                                                                  ⁢                                  C                  N                                                      )                    ⁢                      U            2                    ⁢          sin          ⁢                                          ⁢          θ                                    (        1        )            where σ is the specific gravity of the cable, “CD” is the normal drag coefficient (≈1.5 for a cylindrical cable); CN=0.75CT; and CT=0.0025.
Equation (1) can be solved for θ for any given tow speed. The Strouhal frequency formula (fs=0.2 U sin θ/d) then indicates that the vortex shedding frequency (which is the same as the Strouhal frequency) is nearly constant over a wide speed range (e.g., approximately 6 Hz for a one-inch diameter steel tow cable). This occurs because θ is small enough so that the small angle approximations are valid (i.e., sin θ≈θ, cos θ≈1, and the sin2 θ term can be neglected compared to the sin θ term). As a result, Equation (1) becomes a linear equation in θ, to the first order. When these assumptions are valid, the incidence angle θ becomes approximately
                    θ        ≈                                                            (                                  σ                  -                  1                                )                            ⁢              gd                                      2              ⁢                              C                N                            ⁢                              U                2                                              .                                    (        2        )            
In practice, this result has been verified in sea tests when the strum frequency was measured as a function of tow speed. The 6 Hz transverse waves in the cable then lead to 12 Hz longitudinal waves (because of the frequency doubling effect).
A mile long tow cable that is strumming over an entire length supports a substantial amount of vibration energy (i.e., the vector dot product of force and displacement). In operating tests, such cables have been observed to vibrate with transverse amplitudes of approximately six inches under a cable tension of over one thousand pounds.